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Chapter 6: Q. 22 (page 539)

Approximate the arc length of f (x) on [a, b], using n line segments and the distance formula. Include a sketch of f (x) and the line segments.

fx=ex,a,b=-1,3,n=4

Short Answer

Expert verified

The arc length is 20.6825.

The graph is sketched as below:

Step by step solution

01

Step 1. Given information .

Consider the given functionfx=ex.

02

Step 2. Formula used .

Arc length=limn鈫�鈭�鈭�k=1n1+鈭�yk鈭�x2路鈭�x

03

Step 3. Find arc length .

fx=ex,a=-1,b=3,n=4鈭�x=b-an=3+14=1xk=a+k鈭�xx0=-1+0=-1,x1=-1+1=0x2=-1+2=1,x3=-1+3=2x4=-1+4=3鈭�y1=f0-f1=e0-e-1=1-0.368=0.632鈭�y2=f1-f0=e-1=2.718-1=1.718鈭�y3=f2-f1=e2-e=4.6707鈭�y4=f3-f2=e3-e2=12.6964

Arclength=1+0.399+1+2.951+1+21.8154+1+161.198脳1=1.1827+1.9877+4.7765+12.7856=20.6825

04

Step 4. Sketch of the function fx=ex .

The sketch of the given function is shown below .

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